Theorem subgex 13587

Description: The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)

Assertion

Ref	Expression
subgex	|- ( G e. Grp -> (SubGrp ` G ) e. _V )

Proof of Theorem subgex

Dummy variables s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subg 13581	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2		fveq2 5589	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
3	2	pweqd 3626	. . . 4 |- ( w = G -> ~P ( Base ` w ) = ~P ( Base ` G ) )
4		oveq1 5964	. . . . 5 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
5	4	eleq1d 2275	. . . 4 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
6	3, 5	rabeqbidv 2768	. . 3 |- ( w = G -> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
7		id 19	. . 3 |- ( G e. Grp -> G e. Grp )
8		basfn 12965	. . . . . 6 |- Base Fn _V
9		elex 2785	. . . . . 6 |- ( G e. Grp -> G e. _V )
10		funfvex 5606	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5385	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 414	. . . . 5 |- ( G e. Grp -> ( Base ` G ) e. _V )
13	12	pwexd 4233	. . . 4 |- ( G e. Grp -> ~P ( Base ` G ) e. _V )
14		rabexg 4195	. . . 4 |- ( ~P ( Base ` G ) e. _V -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
15	13, 14	syl 14	. . 3 |- ( G e. Grp -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
16	1, 6, 7, 15	fvmptd3 5686	. 2 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
17	16, 15	eqeltrd 2283	1 |- ( G e. Grp -> (SubGrp ` G ) e. _V )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   {crab 2489   _Vcvv 2773   ~Pcpw 3621   Fn wfn 5275   ` cfv 5280  (class class class)co 5957   Basecbs 12907   ↾_s cress 12908   Grpcgrp 13407  SubGrpcsubg 13578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288  df-ov 5960  df-inn 9057  df-ndx 12910  df-slot 12911  df-base 12913  df-subg 13581

This theorem is referenced by:  isnsg  13613